Download - IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…
![Page 1: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/1.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Harmonic measure of critical curves and CFT Harmonic measure of critical curves and CFT
Ilya A. GruzbergUniversity of Chicago
with
E. Bettelheim, I. Rushkin, and P. Wiegmann
![Page 2: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/2.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
2D critical models2D critical models
Ising model Percolation
![Page 3: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/3.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Critical curvesCritical curves
• Focus on one domain wall using certain boundary conditions
• Conformal invariance: systems in simple domains. Typically, upper half plane
![Page 4: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/4.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Critical curves: geometry and probabilitiesCritical curves: geometry and probabilities
• Fractal dimensions
• Multifractal spectrum of harmonic measure
• Crossing probability
• Left vs. right passage probability
• Many more …
![Page 5: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/5.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Harmonic measure on a curveHarmonic measure on a curve
• Probability that a Brownian particle hits a portion of the curve
• Electrostatic analogy: charge on the portion of the curve (total charge one)
• Related to local behavior of electric field: potential near wedge of angle
![Page 6: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/6.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Harmonic measure on a curveHarmonic measure on a curve
• Electric field of a charged cluster
![Page 7: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/7.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Multifractal exponentsMultifractal exponents
• Lumpy charge distribution on a cluster boundary
• Non-linear is the hallmark of a multifractal
• Problem: find for critical curves
• Cover the curve by small discs of radius
• Charges (probabilities) inside discs
• Moments
![Page 8: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/8.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Conformal multifractalityConformal multifractality
B. Duplantier, 2000
• For critical clusters with central charge
• We obtain this and more using traditional CFT Our method is not restricted to
• Originally obtained by quantum gravity
![Page 9: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/9.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Moments of harmonic measureMoments of harmonic measure
• Global moments
• Local moments
fractal dimension
• Ergodicity
![Page 10: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/10.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Harmonic measure and conformal mapsHarmonic measure and conformal maps
• Harmonic measure is conformally invariant:
• Multifractal spectrum is related to derivative expectation values: connection with SLE.
• Use CFT methods
![Page 11: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/11.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Various uniformizing mapsVarious uniformizing maps
(1) (2)(3) (4)
![Page 12: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/12.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Correlators of boundary operatorsCorrelators of boundary operators
- partition function with modified BC
- boundary condition (BC) changing operator
- partition function
![Page 13: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/13.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Correlators of boundary operatorsCorrelators of boundary operators
• Two step averaging:
1. Average over microscopic degrees of freedom in the presence of a given curve
2. Average over all curves
M. Bauer, D. Bernard
![Page 14: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/14.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Correlators of boundary operatorsCorrelators of boundary operators
• Insert “probes” of harmonic measure: primary operators of dimension
• LHS: fuse
• RHS: statistical independence
• Need only -dependence in the limit
![Page 15: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/15.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Conformal invarianceConformal invariance
• Map exterior of to by that satisfies
• Primary field
• Last factor does not depend on
• Put everything together:
![Page 16: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/16.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Mapping to Coulomb gasMapping to Coulomb gas
• Stat mech models loop models height models Gaussian free field (compactified)
L. Kadanoff, B. Nienhuis, J. Kondev
![Page 17: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/17.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Coulomb gasCoulomb gas
• Parameters
• Phases (similar to SLE)
• Central charge
densedilute
![Page 18: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/18.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Coulomb gas: fields and correlatorsCoulomb gas: fields and correlators
• Vertex “electromagnetic” operators
• Charges
• Holomorphic dimension
• Correlators and neutrality
![Page 19: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/19.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Curve-creating operatorsCurve-creating operators
• Magnetic charge creates a vortex in the field
• To create curves choose
B. Nienhuis
![Page 20: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/20.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Curve-creating operatorsCurve-creating operators
• In traditional CFT notation
- the boundary curve operator is
- the bulk curve operator is
with charge
with charge
![Page 21: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/21.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Multifractal spectrum on the boundaryMultifractal spectrum on the boundary
• One curve on the boundary
• KPZ formula: is the gravitationally dressed dimension!
• The “probe”
![Page 22: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/22.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Generalizations: boundaryGeneralizations: boundary
• Several curves on the boundary
• Higher multifractailty: many curves and points
![Page 23: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/23.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Higher multifractality on the boundaryHigher multifractality on the boundary
• Consider
• Need to find
• Here
![Page 24: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/24.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Higher multifractality on the boundaryHigher multifractality on the boundary
• Write as a two-step average and map to UHP:
• Exponents are dimensions of
primary boundary operators with
• Comparing two expressions for , get
![Page 25: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/25.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Generalizations: bulkGeneralizations: bulk
• Several curves in the bulk
![Page 26: IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure of…](https://reader035.vdocument.in/reader035/viewer/2022062911/5a4d1c127f8b9ab0599f7580/html5/thumbnails/26.jpg)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Open questionsOpen questions
• Spatial structure of harmonic measure on stochastic curves
• Stochastic geometry in critical systems with additional symmetries: Wess-Zumino models, W-algebras, etc.
• Stochastic geometry of growing clusters: DLA, etc: no conformal invariance…
• Prefactor in related to structure constants in CFT